FEM simulations for double diffusive transport mechanism hybrid nano fluid flow in corrugated enclosure by installing uniformly heated and concentrated cylinder

Generation of fluid flow due to simultaneous occurrence of heat and mass diffusions caused by buoyancy differences is termed as double diffusion. Pervasive applications of such diffusion arise in numerous natural and scientific systems. This article investigates double diffusion in naturally convective flow of water-based fluid saturated in corrugated enclosure and containing hybrid nano particles composed of Copper (Cu) and Alumina (Al2O3). Impact of uniformly applied magnetic field is also accounted. To produce thermosolutal convective potential circular cylinder of constant radius is also adjusted by providing uniform temperature and concentration distributions. Finite element approach is capitalized to provide solution of utilized governing equations by utilizing Multiphysics COMSOL software. Wide-range of physical parameters are incorporated to depict their influence on associated distributions (velocity, temperature and concentration). Interesting physical quantities like Nusselt number, Sherwood numbers are also calculated against involved sundry parameters. It is note worthily observed that maximum strength of stream lines \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\psi }_{max})$$\end{document}(ψmax) is 3.3 at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi =0$$\end{document}ϕ=0 and drops to 1.2 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}ϕ is increased to 0.04. Furthermore, in the hydrodynamic case (Ha = 0), it is observed that the velocity field exhibits an increasing trend compared to the hydromagnetic case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left(Ha\ne 0\right),$$\end{document}Ha≠0, which is proved from the attained values of stream-function i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|\psi |}_{max}=11$$\end{document}|ψ|max=11 (in the absence of a magnetic field) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|\psi |}_{max}=3.5$$\end{document}|ψ|max=3.5 (in the presence of a magnetic field). It is revealed from the statistics of Nusselt number that increase in volume fraction of nano particles from 0 to 0.4, heat flux coefficient upsurges up to 7% approximately. Since, present work includes novel physical aspects of thermosolutal diffusion generated due to induction of hybrid nanoparticles in water contained in corrugated enclosure, so this study will provide innovative thought to the researchers to conduct research in this direction.


Mathematical formulation
Geometrical interpretation of work is delineated in Fig. 1 which comprises of square cavity with curved surfaces and circular heated cylindrical obstacle is placed inside the enclosure.Cu-Al 2 O 3 -water hybrid nanofluid.Additionally, an external magnetic field with a strength of (B 0 ) is employed in horizontal direction.Equivalent amounts of copper and alumina-nanoparticles (50% of Cu nanoparticles + 50% of Al 2 O 3 ) is dissolved in water to form hybrid nanofluid.The physical characteristics of a working hybrid nanofluid are assumed to be constant under the Boussinesq assumption with the exception of density variations.

Modelling of problem
Governing equations in dimensional form describing 2D, steady double diffusive natural convective flow of an incompressible electrically conducting hybrid nanofluid are represented as below (see Ref. 36 ) (1) where (u, v) are the components of velocity along the (x, y) , T and p are the temperature and pressure of the hybrid nanofluid.Numerous formulations addressing the thermophysical properties of hybrid nanofluid have been introduced in scientific literature.In this particular study, we have employed the physical parameters for hybrid nanofluid as provided by Takabi and Salehi 36 , which are documented in Table 1.
In terms of dimensions, the associated boundary conditions are represented as follows.
Here, n highlights the normal vector on the boundary.We employ the similarity transformation described in Ref. 37 to transform Eqs.(1-5) along with BCs defined in Eqs.(6-8) into a non-dimensional form.
The dimensionless form of the continuity, momentum, and energy equations are as follows after applying similarity transformation. (3) Table 1.Properties of hybrid nanofluid 36 .

Hybrid-nanofluid characteristics Applied relation
Nanoparticles concentration: Thermal conductivity: Heat capacity: Thermal expansion co-efficient: Thermal diffusivity: The physical parameters implicated in Eqs.(10-14) are defined as follow:

Electrical conductivity:
where the Hartmann number (Ha) illustrates the effects of magnetic forces, Rayleigh number (Ra) demonstrates the impact of buoyancy forces.The local Nusselt numbers, average Nusselt numbers, local Sherwood number and, average Sherwood number at the heated wall of the enclosure can be stated as follows;

Computational procedure
Solution of engineering problems modelled mathematically is attained by two primary approaches i.e. analytical and numerical.Analytical methods provide solution for simple problem because they involve mathematical relations developed for flow situations.On contrary to it, numerical methods have advantages that they provide solution at discrete data points in the domain which are more realistic for implementation in real world processes.Likewise, physical insight of problems in regular domains can be easily accessed through analytical approaches whereas, computational approaches play vital role in attaining feasible outcomes from complexly structured units.Specifically, FEM is typically considered to be the most adaptable and well suited for complicated geometries.For this purpose, COMSOL 38 Multiphysics software based on finite element scheme is used for numerical simulation.COMSOL has many advantages, including its user-friendly interface wide range of capabilities and large user community.However, it also has some disadvantages including its steep learning curve high cost and limited compatibility with other packages.It enables a more accurate discretization of computational domain by distributing it into rectangular and triangular element with execution of hp-refinement.Afterwards, Lagrange interpolation formula is capitalized to produce shape function that defines the behavior of field variables at each node.Currently, quadratic shape functions are used to approximate velocity and temperature fields whereas pressure is estimated by linear shape functions.After discretization of domain, equations at element level are constructed by employing weak formulation and local stiffness matrices are generated which are at last combined to form global matrix for whole domain.Following that, Newton's technique is applied to linearize non-linearized expressions, and the linear system of equations that is produced as a consequence is solved directly by employing elimination-based method solver renowned as PARDISO.Steps involved in the process of computations are shown in Fig. 2. The following convergence condition is established for the iterations using the nonlinear function χ n+1 −χ n χ n+1 < 10 −6 .Figure 3 displays the finer computational grid containing rectangular elements at boundaries and triangular elements in the internal space.Table 2 illustrates the number of elements and degrees of freedom across various refinement levels.

Grid sensitivity test
Examining variation in mesh size, we obtained the numerical solution to assess the influence of mesh size on the results.Five different meshes were examined for this purpose.The average Nusselt number values around the hot inner cylinder was determined for a grid refinement test, as shown in Table 2 and Fig. 4. Table 2 reveals that increasing the number off mesh configurations have minimal effects on the average Nusselt number, which can be considered negligible.Consequently, it is noted that a mesh size consisting of 18,080 elements yields a satisfactory solution for this investigation.3 shows that the results are in good agreement with previous results.To enhance the analysis conducted in this study the developed solver is also validated using the experimental findings of Lizardi et al. 40 regarding natural convection.The validation results are presented in Table 4, demonstrating a favorable agreement between the proposed model and the experimental outcomes.Further validation of the current work is conducted by comparing the experimental and numerical velocity fields as well as the horizontal and vertical velocity distributions presented by Lizardi et al. 40 .A comprehensive analysis presenting agreement between experimental and numerical results for velocity field in the presence and absence of protuberance is compared and displayed in Fig. 5a-i.In addition, Fig. 6a-d reveals a significant

Results and interpretation
In this paper, we investigated magnetically influenced thermosolutal free convection flow of Cu-Al 2 O 3 -H 2 O hybrid nanofluid in a corrugated enclosure emplacing a heated circular cylinder.The enclosure is heated and concentrated from the lower wall, while the corrugated surfaces are kept at a lower temperature and concentration.
In this study, it is crucial to acknowledge that gravitational forces align nearly parallel to the thermal gradient.This sets the stage for a nuanced interplay between the weight of the fluid and buoyancy forces.Unlike scenarios where gravitational forces align parallel to the heat source, the dynamics here involve a distinctive competition.Gravity seeks to impede fluid movement, while buoyancy forces endeavor to instigate circulation, aligned with the temperature gradient.This intricate relationship is extensively explored in our previous work 41,42 .The impact of various physical parameters like Rayleigh number ( 10 4 ≤ Ra ≤ 10 6 ) , Hartmann number (0 ≤ Ha ≤ 100),buoyancy ratio (1 ≤ N ≤ 3) , Lewis number (0.1 ≤ Le ≤ 10) , and hybrid nanofluid volume fraction (0 ≤ φ ≤ 0.05) are examined term of streamlines, isotherms, isoconcentration, average Nusselt number and average Sherwood number.

Effects of Rayleigh number
The impact of Rayleigh number (Ra) on streamlines, isotherms, and isoconcentrations is represented in Fig. 7.
The depicted sketches reveal that by increasing, (Ra) the magnitude of velocity increase.As the (Ra) number increases, the temperature difference between the heated and cold walls also increases, causing density variations in the fluid within the cavity.Consequently, the fluid circulates more strongly, leading to an increase in the vortex within the cavity.It is also observed that the fluid moves up ward near the middle of the cavity due to buoyancy-force caused by the imposed thermal condition, and bifurcates due to the presence of solid cylinder, and then tends to drop near the vicinity of the cold wavy wall.As a result, two primaries symmetric circulations arise beside the centered circular cylinder, one of which is rotated clockwise (indicated by negative sign of values) and the other in the opposite direction.When (Ra) is increased from 10 5 , 10 6 , to 10 7 , the maximum absolute stream function strengthens by 0.39, 2.439, and 12.103 respectively, a similar behaviour is observed in ref. 43 .As the heat source is in the centre of the corrugated and in the lower wall, temperature and concentration patterns are parallel near the heat sources at low ( Ra = 10 5 ), indicating conduction dominant flow.At low (Ra), the effect of buoyancy is weak compared to viscous forces and the streamlines in the fluid tend to be mostly parallel and uniform, As the Ra increases, the buoyancy driven convection becomes dominants and streamlines become curved and start forming convection cell.The increase in (Ra) causes a significant change in isotherms and isoconcentrations.In low Ra regime, the temperature and concentration distribution tend to be relatively uniform and isotherms are approximately parallel to each other.With increasing (Ra), the temperature distribution become highly non-uniform and isotherms are no longer parallel but curve.The velocity profile experiences a slight decrease with an increase in the magnetic field, same behaviour is observed in Ref. 44 .This is attributed to the heightened magnetic field generating Lorentz forces that counteract motion, thereby slowing it down.This observation is depicted in Fig. 8.The gradient of temperature contours decreases minutely with increasing (Ha), indicating that an increase in magnetic field effect leads to a domination of conduction.As a result, the magnetic field has little effect on the distribution of isotherms, and convection becomes less prominent at larger magnetic effects.Since the energy and mass transport equations are identical, the isoconcentration contour exhibit a behaviour similar to that of the isotherms contour, as illustrated in Fig. 8.

Effects of nanoparticle volume fraction
In Fig. 9, the impact of the volume fraction ( φ) of (Cu-Al 2 O 3 -H 2 O) hybrid nanofluid on streamlines, isotherms and isoconcentration is highlighted within the corrugated cavity, considering Pr = 6.2, N = 1, Ra = 10 5 , Le = 2.5 and Ha = 25 .Regardless of (φ) values, the fluid motion exhibits both clockwise and counter-clockwise vortices.With an increase in the volume fraction of nanoparticles the fluid becomes more viscous leading to a reduction in buoyancy force and flow velocity.Consequently, stream function values decrease with the maximum strength of streamlines (ψ max ) dropping from 3.45 at φ = 0 , to 1.25 at φ = 0.04, same behavior is observed in ref. 44 .Negligible changes are occurred in isotherm and isoconcentration distribution for an increase in volume

Concluding remarks
The current study investigates novel physical phenomenon related to thermosolutal diffusion in water, specifically in the presence of (Cu-Al 2 O 3 ) hybrid nanoparticles induced in a corrugated enclosure.The governing equations and boundary conditions are solved using the FEM to obtain a well-defined solution.Some salient outcomes of investigation are itemized as under 1.Convection mode grows as Rayleigh number rises but it diminishes as Lorentz forces rise.2. The rate of heat and mass transmission as well as fluid motion all considerably increase as the Rayleigh number rises.3. The components of fluid motion and transfer of heat are significantly reduced as the magnetic field influence increases.The Hartmann number also has an impact on the structure of the flow circulation.4. It is discovered that the parameters under consideration have a significant impact on the fluid streamlines, isotherms and isoconcentrations behave inside the enclosure.5.The heat transfer rate decreases while the mass transfer rate increases as the Lewis number rises.

Figure 2 .
Figure 2. Flow chart of the FEM.

Figure 3 .
Figure 3. Mesh generation of the wavy cavity.

Figure 5 .
Figure 5. (a-i) Comparison of velocity field of current study with Lizardi et al. 40 .(a) Experimental velocity field without protuberance by Lizardi et al. 40 .(b) Numerical velocity field without protuberance by Lizardi et al. 40 .(c) Numerical velocity field without protuberance, present work.(d) Experimental velocity field with rectangular protuberance by Lizardi et al. 40 .(e) Numerical velocity field with rectangular protuberance by Lizardi et al. 40 .(f) Numerical velocity field with rectangular protuberance, present work.(g) Experimental velocity field with semicircular protuberance by Lizardi et al. 40 .(h) Numerical velocity field with semi-circular protuberance by Lizardi et al. 40 .(i) Numerical velocity field with semi-circular protuberance, present work.

Figure 6 .Figure 8
Figure 6.(a-d) Comparison of vertical and horizontal velocity distribution of current study with Lizardi et al. 40 .(a) Distribution of vertical velocity for the y = 0.04 position by Lizardi et al. 40 .(b) Distribution of vertical velocity for the y = 0.04 position, present work.(c) Distribution of horizontal velocity for the x = 0.04 position by Lizardi et al. 40 .(d) Distribution of horizontal velocity for the x = 0.04 position, present work.

Figure 10
Figure10illustrates the impact of varying the Lewis number (Le) from 0.1 to 10 on momentum, temperature, and concentration distributions when Pr = 6.8,Ha = 25, Ra = 10 4 , N = 1 and ( φ = 0.04 ).As, Lewis number (Le) governs the ratio of thermal diffusivity to mass diffusivity.Therefore, increased Le can be interpreted by the thermal diffusivity's dominance which restricts convective heat transport.It is also found that the maximum absolute stream function is ψ max = 8.7 at Le = 1 and decline to ψ max = 5.2 at Le = 10.Increasing the Lewis num- ber (Le) appears to have no effect on the isotherms.Figure10compares the positive trend in the magnitude of mass flux with respect to the Lewis number (Le).Lewis number define the ratio of thermal to mass diffusivity influences the thermal and mass diffusion characteristics.An increase in (Le) enhances thermal diffusivity but reduces mass diffusivity.As a result, an optimal zone is attained at Le = 0.1, leading to reduced mass dispersion.Meanwhile, a narrower region of isoconcentration is noticeable at Le = 10.Figure11a,b depict the acceleration of the average Nusselt number (Nu avg ) left side and average Sherwood number (Sh avg ) right side in relation to volume fraction of (Cu-Al 2 O 3 -H 2 O) hybrid nanofluid and for various Rayleigh number on the heated circular cylinder and on the heated bottom wall at fixed values of Pr = 6.2, N = 1, Ra = 10 5 and Ha = 25.When Ra is high, free convection flows enhance and heat transfer rate rises.Ra raises the buoyancy force in the enclosure, and convection becomes the major mechanism of heat and mass transport raising Nu avg and Sh avg .

Figure 11 .
Figure 11.Effects of (a) Nu avg and (b) Sh avg for different values of ϕ and Ra.

Figure 12 .
Figure 12.Effects of (a) Nu avg and (b) Sh avg for different values of ϕ and Ha.

Figure 13 .
Figure 13.Effects of (a) Nu avg and (b) SH avg for different values of Ra and N.

Table 3 .
40mparison of current and previous results for the average Nusselt number.resemblancebetweenhorizontal and vertical components of velocity by drawing cutlines in the presence and absence of protuberance.From the displayed sketches, complete agreement between present and existing simulations documented by Lizardi et al.40by both experimental and numerical studies.

Table 4 .
40mparison of current model with experimental data of Lizardi et al.40.